where \(C(x)\)  is independent of \(P\in \mathcal {K}_n\) . Then
taking into account
the asymptotic expansion of \(\phi (P,x)\)  near \(P_0\)  in
(), we obtain (REF ). The representation
() for \(u\)  on \( \widetilde{\Omega }\)  follows from trace
formula (REF ) and the expression (F.59)[1]} for
\(\sum _{j=0}^n \mu _j\) . The representation () for \(\rho \)  on \( \widetilde{\Omega }\) 
is clear from (REF ) and (REF ).
By continuity, (REF ), (), and
() extend from \(\widetilde{\Omega }\)  to \(\Omega \) .
Assuming \(\mu _j \ne 0\) , \(j=0,\ldots ,n\) ,
the constraint (REF ) follows by combining
(REF ), (), and (F.58) [1]}.
Equation (REF ) is clear from
(REF ). Finally, (REF ) and (REF ) extend to
\(\widetilde{\Omega }\)  by continuity.
   \(\square \)
